Easy Algebra Step-by-Step

Factoring Polynomials 135

c. x^2 + 4

d. 8 x^3 − 27

e. 64 a^3 + 125

Solution

a. 9 x^2 − 25 y^2

Step 1. Observe that the binomial has the form “quantity squared minus

quantity squared,” so it is the difference of t wo squares. Indicate that

9 x^2 − 25 y^2 factors as the product of two binomials, one with a plus

sign between the terms and the other with a minus sign between the

terms.

9 x^2 − 25 y^2 = ( + )( − )

Step 2. Fill in the terms of the binomials. The two fi rst terms are the same,

and they both equal 93 xx^23. The two last terms are the same,

and they both equal 25 yy^255.

92 xy^222225555 y =()++ ()− is the factored form.

b. x^2 − 1

Step 1. Observe that the binomial has the form “quantity squared minus

quantity squared,” so it is the difference of two squares. Indicate

that x^2 − 1 factors as the product of two binomials, one with a plus

sign between the terms and the other with a minus sign between the

terms.

x^2 − 1 = ( + )( − )

Step 2. Fill in the terms of the binomials. The two fi rst terms are the same,

and they both equal xx^2. The two last terms are the same, and

they both equal 11.

x^2 −= 11 ()++ )))((()( is the factored form.

c. x^2 + 4

Step 1. Observe that the binomial has the form “quantity

squared plus quantity squared,” so it is the sum

of two squares, and thus is not factorable over the

real numbers.

x^2 + 4 ≠ (x + 2)^2. (x + 2)^2

= x^2 + 4 x + 4, not x^2 + 4.